^{}DEPARTMENT OF MATHEMATICS, BEHBAHAN BRANCH, ISLAMIC AZAD UNIVERSITY, BEHBAHAN, IRAN.

Abstract

Let $A_1$, $A_2$ be unital Banach algebras and $X$ be an $A_1$-$A_2$- module. Applying the concept of module maps, (inner) module generalized derivations and generalized first cohomology groups, we present several results concerning the relations between module generalized derivations from $A_i$ into the dual space $A^*_i$ (for $i=1,2$) and such derivations from the triangular Banach algebra of the form $mathcal{T} :=left(begin{array}{lc} A_1 &X\ 0 & A_2end{array}right)$ into the associated triangular $mathcal{T}$- bimodule $mathcal{T}^*$ of the form $mathcal{T}^*:=left(begin{array}{lc} A_1^* &X^*\ 0 & A_2^*end{array}right)$. In particular, we show that the so-called generalized first cohomology group from $mathcal{T}$ to $mathcal{T}^*$ is isomorphic to the directed sum of the generalized first cohomology group from $A_1$ to $A^*_1$ and the generalized first cohomology group from $A_2$ to $A_2^*$